Models of dynamic systems are of great importance in almost all fields of science and engineering and specifically in control, signal processing, and information science. Most systems encountered in the real world are nonlinear and in many practical applications nonlinear models are required to achieve an adequate modeling accuracy. A model is always only an approximation of a real phenomenon so that having an approximation theory which allows for analysis of model quality is a substantial concern.
A support vector regression (SVR) model was developed for accurate modeling of nonlinear systems. While the SVR capability for representing static nonlinearity has been proven to be powerful in several case studies, its application for nonlinear dynamic system modeling has met several challenges. Particularly, difficulty in implementing the SVR model in parallel configuration for dynamic systems has been identified as one of the key hurdles in this application. Accordingly, there exists a need in the art to provide an SVR model having a parallel or parallel-like configuration for dynamic systems.
The SVR model is optimized and implemented in a series-parallel configuration. FIG. 1A illustrates the series-parallel configuration. The series-parallel configuration takes the measured inputs and outputs to predict the future output of a dynamic system. The model and physical system are running in parallel, but the output of the physical system is fed into the SVR model. For other applications, particularly for cases where outputs are not measured, a model having a parallel configuration, such as shown in FIG. 1B, is required. The parallel model does not rely on output from the physical system to predict the future output. Therefore, it can be run in parallel with the physical system.
While the derivation of the SVR models for the series-parallel implementation can be formulated as a linear programming or quadratic programming optimization problem, the direct optimization of the SVR model for the parallel implementation is computationally prohibitive. Consequently, the derivation of the parallel model is often done as a post-processing step after the series-parallel model as shown in FIG. 1A is derived. It is common for one to derive the series-parallel model using linear programming or quadratic programming and then rewrite the resulting SVR model as shown in FIG. 1A into a parallel implementation as shown in FIG. 1B.
The parallel and series-parallel models are different in structure in that the parallel model requires internal feedback while the series-parallel model does not. The parallel model directly translated from the series-parallel model cannot be guaranteed to have the desired properties, such as stability and model accuracy.